The present invention is related to a method for measuring a thermal expansion coefficient of a thin film by using phase shifting interferometry, and more particularly to a method for measuring a thermal expansion coefficient, intrinsic stress and an elastic modulus of a thin film simultaneously by using phase shifting interferometry.
Tantalum pentoxide (Ta2O5) dielectric film has a high refractive index in the visible region with a wide transmission range extending from 300 nm to about 10 xcexcm. Ta2O5 coatings are widely used in both optical and electronic applications. Some of these applications are as antireflection coatings, optical waveguides, metal oxide semiconductor (MOS) devices, insulator in electronic devices, and as narrow-bandpass filters. The temperature stability of optical coatings has become more and more important, especially for optical telecommunications. Narrow-bandpass filters (NBF) are the key components in the elimination of noise in fibre amplifiers and for wavelength selection in high-density wavelength-division-multiplexed systems. One of the key design parameters in the NBF is the coefficient of thermal expansion (CTE). In general, thermal effects provide important contributions to film stress. Films prepared at an elevated temperature and cooled to room temperature will be thermally strained.
For a thin film deposited on a substrate, a stress of the thin film will cause the substrate deflect downward or upward. The deflection is downward for a tensile stress and upward for a compressive stress. In either case, the thin film might detach from the substrate when the stress is too large. The stress of the thin film is composed of two components, which are intrinsic stress, "sgr"i, and thermal stress, "sgr"T, if no external stress is exerted thereon. The intrinsic stress is a result of interaction between the growth modes and the microstructure of the thin film, and the thermal stress is caused by different values of thermal expansion coefficients between the thin film and the substrate. The thermal stress can be represented by the following formula:       σ    T    =            (                        α          s                -                  α          f                    )        ⁢                  E        f                    (                  1          -                      v            f                          )              ⁢          (                        T          2                -                  T          1                    )      
wherein xcex1s is the thermal expansion coefficient of the substrate, xcex1f and Ef are the thermal expansion coefficient and Young""s modulus of the thin film, respectively, xcexdf is Poisson""s ratio of the thin film, T1 is a deposition temperature of the thin film. It can be understood from the above formula, the thermal stress, "sgr"T, at a measuring temperature, T2, can be calculated if xcex1s, xcex1f and       E    f        (          1      -              v        f              )  
are known. The stress of the thin film at measuring temperature, T2, can also be obtained if the intrinsic stress, "sgr"i, is known. Briefly, the xcex1f and       E    f        (          1      -              v        f              )  
so measured will enable a person calculate, in advance, a stress of the thin film deposited on the substrate at a pre-determined temperature.
A number of techniques for measuring the thermal expansion coefficients of thin films have been developed, such as the interference fringe method, the optical levered laser technique, the bending beam technique and the capacitance cell method. In these techniques, the principle of measurement is detecting the deflections caused by the stress of the thin film at different temperatures, and calculating th hermal expansion coefficient, intrinsic stress and elastic modulus,             E      f              (              1        -                  v          f                    )        ,
by using the relationship between the stress of the thin film and the temperatures. One suitable method for measuring the deflections of the thin film deposited on a substrate is the interferometric technique [A.E. Ennos, xe2x80x9cStress developed in optical film coatingsxe2x80x9d, Appl. Opt. Vol 5, No. 1, pp.51-61, 1966; K. Roll and H. Hoffmann, xe2x80x9cMichelson interferometer for deformation measurements in an UHV system at elevated temperaturesxe2x80x9d, Rev. Sci. Instrum., Vol 47, No. 9, pp.1183-1185, 1976.] The above interferometric technique is somewhat elaborate, and inaccurate because the difference of the number of fringe in two fringe patterns is required to be an integer.
The present invention disclose a method for measuring a thermal expansion coefficient of a thin film, in which the thin film is first deposited on two substrates having different thermal expansion coefficients under the same conditions. For each of the two deposited substrates, a relationship between the thin film stresses and the measuring temperatures is established by using a phase shifting interferometry technique, in which the stresses in the thin films are derived by comparing the deflections of the substrates prior to and after the deposition. Based the two relationships the intrinsic stress, thermal expansion coefficient, and elastic modulus,             E      f              (              1        -                  v          f                    )        ,
can be calculated. Alternatively, the stresses of the thin films deposited on two different substrates are plotted against the stress measuring temperatures, showing a linear dependence. From the slopes of the two lines in the stress versus temperature plot, the intrinsic stress, thermal expansion coefficient and elastic modulus of the thin film is determined, simultaneously. The present method is relatively simple and convenient and can be extended to varying-temperature applications without damaging the thin film.
The method for measuring a thermal expansion coefficient of a thin film deposited on a substrate by phase shifting interferometry accomplished in accordance with the present invention comprises the following steps:
a) measuring a phase function of a target surface of a first substrate at a first measuring temperature;
b) depositing a thin film on said target surface of said first substrate;
c) measuring a phase function of said thin film by using the same conditions as those in step a);
d) calculating one or more relative heights of one or more points with respect to a central point of said substrate prior to and after the deposition in step b) by using the phase functions obtained in steps a) and c), respectively, and calculating a difference of the relative heights at a same point prior to and after the deposition in step b) for each of said one or more points;
e) calculating a stress of said thin film for each of said one or more points by using said difference from step d) and calculating an average stress of said thin film therefrom;
f) obtaining another one or more average stresses of said thin film of another one or more measuring temperatures by repeating steps a), c), d) and e) except that said first measuring temperature is replaced by said another one or more temperatures, and obtaining a first set of data of said average stresses versus said measuring temperatures with respect to said first substrate;
g) measuring a phase function of a target surface of a second substrate at a second measuring temperature, wherein the second substrate has a thermal expansion coefficient different from that of said first substrate;
h) depositing a thin film on said target surface of said second substrate with the conditions same as those in step b);
i) measuring a phase function of said thin film in step h) by using the same conditions as those in step g);
j) calculating one or more relative heights of one or more points with respect to a central point of said substrate prior to and after the deposition in step h) by using the phase functions obtained in steps g) and i), respectively, and calculating a difference of the relative heights at a same point prior to and after the deposition in step h) for each of said one or more points;
k) calculating a stress of said thin film of step h) for each of said one or more points by using said difference from step j) and calculating an average stress of said thin film of step h) therefrom;
l) obtaining another one or more average stresses of said thin film of step h) of another one or more measuring temperatures by repeating steps g), i), j) and k) except that said second measuring temperature is replaced by said another one or more temperatures, and obtaining a second set of data of said average stresses versus said measuring temperatures with respect to said second substrate; and
m) calculating a thermal expansion coefficient, xcex1f, of said thin film by using said first set data and said second set data.
Preferably, said thermal expansion coefficient, xcex1f, is calculated according to the following formula (1):                     σ        =                              σ            i                    +                                    (                                                α                  s                                -                                  α                  f                                            )                        ⁢                                          E                f                                            (                                  1                  -                                      v                    f                                                  )                                      ⁢                          (                                                T                  2                                -                                  T                  1                                            )                                                          (        1        )            
wherein T2 is any one said measuring temperatures, "sgr" is said average stress of said thin film at T2, "sgr"i is an intrinsic stress of said thin film, xcex1s is a thermal expansion coefficient of said first substrate or said second substrate, Ef is a Young""s modulus of said thin film, xcexdf is a Poisson""s ratio of said thin film, T1 is a temperature at which said thin film is deposited in step b) and step h).
In the method of present invention, preferably, an elastic modulus defined as follows is obtained in step m) together with said thermal expansion coefficient, xcex1f:       E    f        (          1      -              v        f              )  
wherein Ef and xcexdf are defined as above.
Preferably, said intrinsic stress, "sgr"i, of said thin film is obtained in step m) together with said thermal expansion coefficient, xcex1f. More preferably, one of said measuring temperatures (T2) is chosen to be equal to said temperature (T1) at which said thin film is deposited in step b) and step h), so that said intrinsic stress, "sgr"i, is equal to said average stress of said thin film at T2, "sgr", according to the formula (1).
Preferably, said measurement of phase function in step a), c), g) and i) comprises the following steps:
I) generating two reflection light beams from a reference plate and a target surface of a substrate by perpendicularly irradiating two light beams splitted from one light source on said reference plate and said target surface;
II) recombining said two reflection light beams into one single beam and directing said one single beam on a screen to form an interference pattern;
III) digitizing said interference pattern to obtain digitized light intensities thereof;
IV) moving said reference plate or said substrate to several positions in a direction parallel to said light beam irradiating thereon, and said several positions being equally spaced from an original position of said reference plate or said substrate;
V) repeating steps I) to ll) to obtain digitized light intensities of interference pattern for each one of said several positions; and
VI) calculating a phase function of said target surface of said substrate by using said light intensities of said interference patterns obtained from step III) and step V).
Preferably, said reference plate is moved to said several positions in step IV) by using a piezoelectric transducer (PZT) connected to a surface of said reference plate opposite to said light beam irradiating thereon, and by using a DC power supply controlled by a computer to supply a pre-determined voltage to said PZT.
Preferably, said digitizing said interference pattern to obtain light intensity thereof in step III) comprises forming a image of said interference pattern by using a CCD camera and digitizing a light intensity of each pixel of said image.
Preferably, said digitized light intensities are provided to said computer, said phase functions in steps a), c), g) and i), said relative heights and said differences in steps d) and j), and said stress and said average stress in steps e) and k) are calculated by using said computer in associated with programs stored therein.
Preferably, said reference plate is moved to four positions in step IV) which are equally spaced of xcex/8 from an original position of said reference plate, wherein xcex is a wavelength of said irradiating light beam on said reference plate, so that five set of light intensities of said interference patterns in steps III) and V) representing 0xc2x0, 90xc2x0, 180xc2x0, 270xc2x0, and 360xc2x0 phase shifts are obtained, and said phase function is calculated by the following formula:   Φ  =            tan              -        1              ⁡          [                        2          ⁢                      (                                          I                2                            -                              I                4                                      )                                                2            ⁢                          I              3                                -                      I            5                    -                      I            1                              ]      
wherein I1, I2, I3, I4 and I5 represent said digitized light intensities at a particular pixel of the five interference patterns, and "PHgr" is a phase of said particular pixel.
Preferably, said relative heights in steps d) and j) are calculated according to the following formula:       relative    ⁢          xe2x80x83        ⁢    height    =            0.6328              4        ⁢        π              ⁢          (                        Φ          r                -                  Φ          c                    )      
wherein "PHgr"r is a phase of a point on said substrate at a radius r from said central point thereof, and
"PHgr"c is a phase of said central point of said substrate.
Preferably, said light source in step l) is a laser light or a slit light source generated by passing a light through a slit.
Preferably, the method of the present invention may further comprises repeating steps g) to m) for one or more substrates having different thermal expansion coefficients different from those of the first and second substrates, so that additional sets of data of said average stresses versus said measuring temperatures with respect to said one or more substrates are obtained, and so that said thermal expansion coefficient, xcex1f, of said thin film can be calculated by using said additional sets of data together with said first set data and said second set data.